1) The document discusses finding measures of angles formed by chords, secants, and tangents of circles. It provides theorems for calculating angle measures based on intercepted arc lengths. 2) Formulas are presented for calculating lengths of segments associated with circles, including secants, chords, and tangents. 3) Examples demonstrate using the theorems and formulas to find missing angle measures and segment lengths in various circle geometry problems involving combinations of secants, chords and tangents.

1. Sec. 14-5Sec. 14-5
∠∠ Measures & SegmentMeasures & Segment
Lengths in CirclesLengths in Circles
Objectives:Objectives:
1) To find the measures of1) To find the measures of ∠∠s formeds formed
by chords, secants, & tangents.by chords, secants, & tangents.
2) To find the lengths of segments2) To find the lengths of segments
associated with circles.associated with circles.

2. SecantsSecants
E
A
B
F
Secant – A line that
intersects a circle in exactly
2 points.
•EF or AB are secants
•AB is a chord

3.  (Thm 11 – 11) The measure of an(Thm 11 – 11) The measure of an ∠∠
formed by 2 lines that intersectformed by 2 lines that intersect insideinside aa
circle iscircle is
m∠1 = ½(x + y)
Measure of intercepted arcs
1
x°
y°

4.  (…Thm 11 – 11 Continues) The measure(…Thm 11 – 11 Continues) The measure
of anof an ∠∠ formed by 2 lines that intersectformed by 2 lines that intersect
outsideoutside a circle isa circle is
m∠1 = ½(x - y) Smaller Arc
Larger Arc
x°
y°
1
x°
y°
1
2 Secants:
x°
y°
1
Tangent & a Secant
2 Tangents
3 cases:

5. Ex.1 & 2:Ex.1 & 2:
 Find the measure ofFind the measure of
arc x.arc x.
 Find the mFind the m∠∠x.x.
94°
112°
x°
m∠1 = ½(x + y)
94 = ½(112 + x)
188 = (112 + x)
76° = x
68° 104°
92°
268°
x°
m∠x = ½(x - y)
m∠x = ½(268 - 92)
m∠x = ½(176)
m∠x = 88°

6. Thm (11 – 12) Lengths of Secants,Thm (11 – 12) Lengths of Secants,
Tangents, & ChordsTangents, & Chords
2 Chords
a c
b
d
a•b = c•d
2 Secants
x
w
z
y
w(w + x) = y(y + z)
Tangent & Secant
t
y
z
t2
= y(y + z)

7. Ex. 3 & 4Ex. 3 & 4
 Find length of x.Find length of x.
 Find the length of g.Find the length of g.
3 x
7
5
15
8
g

8. Ex. 3 & 4Ex. 3 & 4
 Find length of x.Find length of x.
 Find the length of g.Find the length of g.
3 x
7
5
a•b = c•d
(3)•(7) = (x)•(5)
21 = 5x
4.2 = x
15
8
g
t2
= y(y + z)
152
= 8(8 + g)
225 = 64 + 8g
161 = 8g
20.125 = g

9. Ex.5: 2 SecantsEx.5: 2 Secants
 Find the length of x.Find the length of x.
14
20
16
x
w(w + x) = y(y + z)
14(14 + 20) = 16(16 + x)
(34)(14) = 256 + 16x
476 = 256 + 16x
220 = 16x
3.75 = x

10. Ex.6: A little bit of everything!Ex.6: A little bit of everything!
 Find the measures of the missing variablesFind the measures of the missing variables
9
12
k
8
a°
r
60°
175°

11. Ex.6: A little bit of everything!Ex.6: A little bit of everything!
 Find the measures of the missing variablesFind the measures of the missing variables
9
12
k
8
a°
r
60°
175°
Solve for k first.
w(w + x) = y(y + z)
9(9 + 12) = 8(8 + k)
186 = 64 + 8k
k = 15.6
Next solve for r
t2
= y(y + z)
r2
= 8(8 + 15.6)
r2
= 189
r = 13.7
Lastly solve for
m∠a
m∠1 = ½(x - y)
m∠a = ½(175 – 60)

12. What have we learned??What have we learned??
 When dealing with angle measures formed byWhen dealing with angle measures formed by
intersecting secants or tangents you either addintersecting secants or tangents you either add
or subtract the intercepted arcs depending onor subtract the intercepted arcs depending on
where the lines intersect.where the lines intersect.
 There are 3 formulas to solve for segmentsThere are 3 formulas to solve for segments
lengths inside of circles, it depends on whichlengths inside of circles, it depends on which
segments you are dealing with: Secants,segments you are dealing with: Secants,
Chords, or Tangents.Chords, or Tangents.